Carnot's theoremtheorem

In 1824, French engineer Sadi Carnot, pondering the improvement of steam engines, arrived at a fundamental theorem. He realized that the maximum efficiency of any heat engine is determined solely by the temperatures of the heater and the cooler and does not depend on the working substance. The idea came from an analogy with a water mill: the work is determined not by the mechanism but by the difference in levels. Carnot introduced the concept of a reversible cycle where there are no losses due to friction, etc., and proved that all reversible engines have the same efficiency, and irreversible ones have less. This result later formed the basis of the second law of thermodynamics.

Carnot's theorem states: no engine can be more efficient than an ideal one operating on the Carnot cycle. All the heat that a system receives from a hot source (e.g., combustion) cannot be completely turned into useful work—some always goes to the cold source (the environment). This is like a water wheel: water from a high level (high temperature) flows down to a low one (low temperature), and the maximum work depends on the height difference, not on the wheel's design.

Carnot's theorem states that the efficiency of any heat engine operating on a reversible or irreversible cycle between two heat reservoirs at temperatures T₁ (heater) and T₂ (cooler, T₂ < T₁) cannot exceed the efficiency of a reversible Carnot cycle: η_max = 1 − T₂/T₁. A consequence is the impossibility of a perpetual motion machine of the second kind. A rigorous proof uses the second law of thermodynamics in Clausius's formulation. The theorem sets a fundamental limit for the conversion of thermal energy into work.